Tính giới hạn \(\mathop {\lim }\limits_{n \to  + \infty } \frac{{1 + 2 + 3 + ... + \left( {n - 1} \right) + n}}{{{n^2}}}\) 

* Hướng dẫn giải

Ta có: \(\mathop {\lim }\limits_{x \to \infty } \frac{{1 + 2 + 3 + ... + \left( {n - 1} \right) + n}}{{{n^2}}} = \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{{n\left( {n + 1} \right)}}{2}}}{{{n^2}}} = \mathop {\lim }\limits_{x \to \infty } \frac{{{n^2} + n}}{{2{n^2}}} = \frac{1}{2}\)